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A306387
Number of partitions of sigma_1(n) into divisors of n.
0
1, 2, 2, 6, 2, 27, 2, 26, 7, 31, 2, 574, 2, 38, 33, 166, 2, 879, 2, 924, 39, 52, 2, 23732, 9, 59, 47, 1403, 2, 34256, 2, 1626, 55, 73, 47, 230819, 2, 80, 61, 50888, 2, 65638, 2, 2709, 1734, 94, 2, 2117920, 11, 3038, 77, 3536, 2, 113448, 65, 97298, 83, 115, 2, 19613170, 2, 122, 2601, 25510, 73, 180350
OFFSET
1,2
COMMENTS
The equality sigma_1(n) = Sum{d|n} d defines one partition of sigma_1(n) into distinct divisors of n. This sequence gives the number of partitions of sigma_1(n) into not necessarily distinct divisors of n.
For prime number p, sigma_1(p) = p+1 and there are only two partitions: p and 1+1+1+...+1 (p summands).
EXAMPLE
For n = 4, sigma_1(4) = 7, divisors(4) = {1,2,4} and 7 = 4+2+1 = 4+1+1+1 = 2+2+2+1 = 2+2+1+1+1 = 2+1+1+1+1+1 = 1+1+1+1+1+1+1.
For n = 9, sigma_1(9) = 13, divisors(9) = {1,3,9} and 13 = 9+3+1 = 9+1+1+1+1 = 3+3+3+3+1 = 3+3+3+1+1+1+1 = 3+3+1+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1+1.
PROG
(Magma) v:=[1..47];
for u in v do
u, #RestrictedPartitions(SumOfDivisors(u), {d:d in Divisors(u)});
end for;
(Magma)
a:= func< n | #RestrictedPartitions(SumOfDivisors(n), {d:d in Divisors(n)}) >; [ a(n) : n in [1..47] ];
(PARI) numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1, mx, numbpartUsing(n-v[i], v, i)) \\ inefficient;
a(n) = numbpartUsing(sigma(n), divisors(n)); \\ after A018818; Michel Marcus, Feb 27 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Feb 26 2019
STATUS
approved